SLALM 2024, the 20th Latin American Symposium on Mathematical Logic of the Association of Symbolic Logic was hosted by the Facultad de Ingeniería at the Universidad de la República, Montevideo, Uruguay, July 1 through July 5, 2024. The conference was attended by a total of 86 participants. The local organizing committee was coordinated by Mauricio Guillermo and consisted of Walter Ferrer, Juan García Garland, Octavio Malherbe, and Alberto Pardo.

Funding for the conference was provided by the Universidad de la República, including the Facultad de Ingeniería, the Instituto de Matmática y Estadística Rafael Laguardia, the Centro de Matemática, the Centro Universitario Regional del Este, and the Comisión Sectorial de Investigación Cientifica, as well as by the Association for Symbolic Logic, PEDECIBA (Programa de Desarrollo de las Ciencias Basicas) of Uruguay, and the National Science Foundation of the United States.

The Program Committee was chaired by Antonio Montalbán (University of California Berkeley) and consisted of Christina Brech (Universidade de São Paulo), Manuela Busaniche (Universidad Nacional del Litoral), Andrés Caicedo (University of Michigan), Xavier Caicedo Ferrer (Universidad de los Andes), Russell G. Miller (City University of New York), Lourdes del Carmen González Huesca (Universidad Nacional Autónoma de México), Deirdre Haskell (McMaster University), Octavio Malherbe (Universidad de la República), Alf Onshuus (Universidad de los Andes), Hector Pasten (Pontificia Universidad Católica de Chile), Luiz Carlos Pereira (Pontificia Universidade Católica do Rio de Janeiro), and Elaine Pimentel (University College London).

The program included three tutorial courses, eight plenary lectures, fifteen invited lectures in five special sessions (logic in computer science and computability, model theory, non-classical logics, philosophical logic, and set theory), and 43 contributed talks. The following tutorial courses were given.

Natasha Dobrinen (University of Notre Dame), Infinite structural Ramsey theory and logic.

Nadja Hempel (Universität Düsseldorf), A tutorial on the tame world of groups and fields.

Marcos Silva (Universidade Federal de Pernambuco), How to revise logic from a neopragmatist point of view.

The following invited plenary lectures were presented.

Pierre-Louis Curien (CNRS, Université Paris Cité), Clones as monoids.

Carlos Di Prisco (Universidad de los Andes), Combinatorial properties of ideals.

Alejandro Díaz-Caro (Universidad de Buenos Aires and Universidad Nacional de Quilmes), From linear logic to quantum control.

Itala D’Ottaviano (Universidade de Campinas), Charles Sanders Peirce and the horizons of logic for the 20th century.

Alexandre Miquel (Universidad de la República), Implicative algebras: a survey

Samaria Montenegro (Universidad de Costa Rica), Fields and NIP2.

Davide Sangiorgi (Università di Bologna), Proof techniques for behavioral relations: from enhanced coinduction to enhanced induction

Peter Selinger (Dalhousie University), Quantum computing from a logician’s point of view.

Abstracts of invited and contributed talks given in person or by title by members of the Association follow.

For the Program Committee

Antonio Montalbán

Abstract of invited tutorials

▸NATASHA DOBRINEN, Infinite structural Ramsey theory and logic.

Department of Mathematics, University of Notre Dame, Notre Dame, Indiana, USA.

E-mail: ndobrine@nd.edu.

The infinite Ramsey theorem states that given any coloring of all pairs of natural numbers into two colors, there is an infinite subset of natural numbers in which all pairs have the same color. When moving from sets to relational structures, some surprising phenomena occur: The prototypical example is that there is a coloring of pairs of rational numbers into two colors such that both colors persist in any subset of the rationals forming a dense linear order (Sierpínski, 1933). Likewise for colorings of edges in the Rado graph (Erdős–Hajnal–Pósa, 1975). The study of optimal bounds for finite colorings of copies (or embeddings) of a finite substructure inside an infinite structure is the subject of big Ramsey degrees. Optimal bounds are connected with structural expansions which produce analogs of the infinite Ramsey theorem; the pursuit of the optimal structural expansions has led to new connections between logic and structural Ramsey theory.

This tutorial will introduce big Ramsey degrees, key examples, and components intrinsic to their characterizations. We will examine proof methods, including Milliken’s strong tree theorem, Harrington’s forcing proof of the Halpern–Läuchli theorem, coding trees and forcing Ramsey theorems on them, parameter words, and others. A natural extension of big Ramsey degrees to infinite-dimensional structural Ramsey theory ties in with topological Ramsey spaces. We will cover the basics of this area as well as recent work on Ramsey spaces of infinite structures. By the end of the tutorial, we will have given the big picture of current knowledge in the area, methods, and ongoing work.

The motivation for and progress of Ramsey theory on infinite structures are intrinsically intertwined with problems and methods in logic, including first-order logic, set theory, model theory, and computability theory. These connections will be exhibited throughout the tutorial. The paper [1] provides a gentle introduction to infinite structural Ramsey theory and an overiew of the area, making it a good companion to the tutorial. References from very recently published papers will be included throughout the tutorial.

[1] N. Dobrinen, Ramsey theory of homogeneous structures: Current trends and open problems, ICM—International Congress of Mathematicians, Volume 3, Sections 1–4 (D. Beliaev and S. Smirnov, editors), European Mathematical Society, Berlin, 2023, pp. 1462–1486.

▸NADJA HEMPEL, Model theory: a tutorial on the tame world of groups and field.

Mathematisches Institut, Heinrich Heine Universität, Düsseldorf, Germany.

E-mail: nadja.valentin@hhu.de.

This Model Theory Tutorial is designed for logicians with limited mathematical and model theoretic background.

Model theory is a branch of mathematical logic that studies structures like graphs, groups, and fields through formal logical languages, particularly first-order logic. It allows results to be transferred between different structures and identifies common features across various mathematical frameworks, thereby providing a unified approach to algebraic, analytic, differential, and difference geometry. A key goal in model theory is distinguishing between tame structures (e.g., the complex field) and wild structures (e.g., the ring of integers). This distinction is based on combinatorial notions of tameness identified by Shelah in the 1970s (Shelah’s Classification Theory).

In the first talk, we provide a gentle introduction, using examples from groups and fields to illustrate the various concepts under consideration. We present the different classes in Shelah’s (binary) Classification Theory: At the apex are the stable structures, such as the complex field, with two complementary extensions: NIP structures (including p-adic fields) and simple structures (such as pseudofinite fields). Additionally, there is a common generalization of NIP and simple structures, namely NTP $_2$ structures. We give examples and discuss properties of groups whose theory falls within these classes.

In the second talk, we attempt to classify fields whose theories belong to one of the classes presented in the first talk.

In the last talk, we leave the binary world, and introduce “n-ary” Classification theory: NIP (also known as 1-dependent) mark the first level of the so-called n-dependent theories. Instead of “controlling” 2-ary relation, we only assume that n-ary relations are tame. We give examples and discuss groups and fields within these classes of theories.

▸MARCOS SILVA, How to revise logic from a neopragmatist point of view.

Universidade Federal de Pernambuco and CNPq, Recife, Brazil.

E-mail: marcossilvarj@gmail.com.

In this tutorial we will present and discuss philosophical and epistemological problems concerning the revision of logic, as circularity, infinite regress, and arbitrariness. Then, we will introduce some methods in a neopragmatist tradition to revise logics. We will draw some lessons from enactivism, semantic inferentialism, and logical expressivism.

Day 1: Revising Logic: On Neo Pragmatism, Inferentialism, and Expressivism

Day 2: On Verbal Disputes and Metalinguistic Negotiations

Day 3: On Normative Affordances and Inferential Abilities

Abstracts of invited plenary lectures

▸PIERRE-LOUIS CURIEN. Clones as monoids.

IRIF and Picube team, CNRS, Université Paris Cité and Inria, France.

E-mail: curien@irif.fr.

Clones (in concrete form or in abstract form) are a fundamental tool in universal algebra and are (in the abstract form) the counterpart in that community of Lawvere theories in the categorical logic community. In recent works, going through infinitary operations, Bucciarelli and Salibra introduced the closely related notion of clone algebras (which unlike clones are mono-sorted). In this talk we exhibit those as certain monoids with an additional structure called merge algebra structure (that abstracts away properties of merged sequences).

This work is joint work with Antonio Bucciarelli and Antonino Salibra.

▸CARLOS A. DI PRISCO. Combinatorial properties of ideals.

Instituto Venezolano de Investigaciones Científicas, Universidad Central de Venezuela and Universidad de Los Andes.

E-mail: cdiprisc@gmail.com.

An ideal on a set A is a collection $\mathcal I$ of subsets of A closed under subsets and finite unions.

Given an ideal $\mathcal I$ over A, a set $B\subseteq A$ is said to be $\mathcal I$ -positive if it does not belong to the ideal. The coideal associated to $\mathcal I$ is the collection $\mathcal I^+=\{B\subseteq A: B\notin \mathcal I\}$ of $\mathcal I$ -positive sets.

We will examine several combinatorial properties of ideals on the set of natural numbers $\mathbb N$ and explore how they relate to each other.

Ramsey’s theorem shows that for every partition of $[\mathbb N]^2$ , the set of two-element subsets of $\mathbb N$ , in finitely many parts, there is an infinite set $H\subseteq \mathbb N$ with all its two-element subsets in one of the parts. In some occasions we want to find such a set H in a coideal, and this motivates the main definition of this talk: An ideal on $\mathbb N$ is Ramsey if for every $A\in \mathcal I^+$ and every partition of $[A]^2$ in two parts, there is an $\mathcal I$ -positive $B\subseteq A$ such that $[B]^2$ is contained in one of the parts. Ramsey ideals and ideals with other combinatorial properties provide a rich and active field of research.

[1] M. Hrusak, Combinatorics of filters and ideals. Contemporary Mathematics , vol. 533 (2011), pp. 29–69.

[2] C. Uzcátegui, Ideals on countable sets: A survey with questions. Revista Integración, Universidad Industrial de Santander , vol. 37 (2019), no. 1, pp. 167–198.

▸ALEJANDRO DÍAZ-CARO, From linear logic to quantum control.

CONICET, Instituto de Ciencias de la Computación, Universidad de Buenos Aires, and Departamento de Ciencia y Tecnología, Universidad Nacional de Quilmes, Buenos Aires, Argentina.

E-mail: alejandro@diaz-caro.info.

This talk explores several recent papers with Gilles Dowek and Octavio Malherbe in the quest for a quantum logic founded on the Curry–Howard–Lambek correspondence. We operate within the quantum control paradigm, where the control flow of programs can follow a quantum particle. Our starting point is intuitionistic multiplicative additive linear logic (IMALL) with an extended proof language, the $L^S$ -calculus, which allows for the representation of matrices, vectors, and measurement-free quantum programs. We then introduce a new logical connective, sup, to express superposition and measurement. Next, we model a probabilistic version of this proof language for IMALL in a symmetric monoidal closed category with biproducts, provided that there is a monomorphism from the set of scalars to the set Hom(I,I). Finally, we present an alternative approach where the sup connective can be avoided by adding an extra introduction rule to the additive disjunction.

▸ITALA M. LOFFREDO D’OTTAVIANO, Charles Sanders Peirce and the horizons of logic for the 20th century.

Centre for Logic, Epistemology and the History of Science, Philosophy Department, University of Campinas, Campinas, Brazil.

E-mail: itala@unicamp.br.

Charles Sanders Peirce introduced the method of truth tables in 1885, in his paper “On the algebra of logic: a contribution to the philosophy of notation”, decades before Russell and Wittgenstein.

In a manuscript in his known Logic Notebook (1865–1909), dated February 23, 1909, and unpublished until 1966, Peirce presented in matrix form the logical basis for a three-valued propositional system, anticipating by more than ten years the many-valued systems introduced in 1920 by Ján Łukasiewicz, and Emil Post in 1921.

In this presentation, we will analyze this manuscript, in the context of Peirce’s intellectual production, making claims about his pioneerism in the usage of concepts of contemporary logic and in the usage of truth tables, and placing him among the forerunners of non-classical logics in general, in particular many-valued, intuitionistic and paraconsistent logics, introduced in the 20th century.

▸ALEXANDRE MIQUEL, Implicative algebras: a survey.

IMERL, Facultad de Ingeniería, Universidad de la República, Montevideo, Uruguay.

E-mail: amiquel@fing.edu.uy.

Implicative algebras are a generalization of complete Heyting algebras intended to factorize the model-theoretic constructions underlying forcing and realizability, both in intuitionistic and classical logic. In this talk, I will first recall the definition of this structure, that relies on the identification of the notion of truth value with the notion of realizer. Then, I will show that with the suitable notion of morphism, implicative algebras form a locally small category that is equivalent to (and thus can be used to represent) a particular category of functors, namely: the category of Set-based triposes (in the sense of Hyland, Johnstone and Pitts) equipped with finite meet-preserving natural transformations. Regarding set theory, I will show that implicative algebras can also be used to construct models of (intuitionistic and classical) set theory, that encompass Boolean-valued models of ZF as well as intuitionistic and classical realizability models of (I)ZF. I will conclude by presenting some perspectives of using implicative algebras to interpret first-order theories.

▸SAMARIA MONTENEGRO, Fields and NTP2.

Escuela de Matemática, Universidad de Costa Rica, San José, Costa Rica.

E-mail: samaria.montenegroguzman@ucr.ac.cr.

One of the objects of study of model theory are the complete first order theories and their classification. Shelah classified complete first order theories by their ability to encode certain combinatorial configurations. The importance of Shelah’s classification theory lies in its ability to provide tools and conceptual frameworks for understanding the structure of mathematical theories and for addressing a wide range of problems in model theory and related areas. His work has influenced numerous subsequent developments in the field and has significantly contributed to advancing knowledge in model theory and related disciplines.

In this talk we are going to focus on the class of NTP $_2$ theories (theories without the tree property of the second kind), this class contains strictly the class of simple and NIP theories. In particular we want to study what type of fields are NTP $_2$ . We already know that the PAC, PRC, and PpC bounded fields are NTP $_2$ , we propose a unified framework for studying these fields—the class of pseudo-T-closed fields, where T is an enriched theory of fields. These fields verify a “local-global” principle for the existence of points on varieties based on models of T. This approach also enables a good description of some fields equipped with multiple V-topologies, particularly pseudo-algebraically closed fields with a finite number of V-topologies. We are going to show how we can use this approach to produce many new examples of NTP $_2$ fields. This talk is a joint work with Silvain Rideau-Kikuchi.

▸DAVIDE SANGIORGI, Proof techniques for behavioural relations: from enhanced coinduction to enhanced induction.

Department of Computer Science, University of Bologna, Bologna, Italy.

E-mail: davide.sangiorgi@gmail.com.

Enhancements of the coinduction proof method—the so-called ‘up-to techniques’—are widely used on coinductive behavioural relations such as bisimilarity. Sometimes these techniques represent in fact the main reason for choosing bisimilarity as a behavioural relation. Bisimilarity, however, can sometimes be considered too discriminating. Moreover, it does not have a natural associated preorder. For these reasons, a number of ‘inductive’ behaviour relations, i.e., relations defined from inductive observables, have put forward and studied since the early days of concurrency theory.

We show how to transport the ‘up-to techniques’ onto the inductive behavioural relations. We then propose an abstract account of the theory, using fixed-point theory in complete lattices.

▸PETER SELINGER, Quantum computing from a logician’s point of view.

Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada.

E-mail: selinger@mathstat.dal.ca.

One may say that logic is concerned with syntax and models. But what is quantum theory concerned with? Ten people will give you ten different answers to this question. Physicists might talk about slit experiments and Schrödinger’s equation, mathematicians about operator algebras, chemists about spin orbitals, computer scientists about qubits and circuits, and philosophers about cats that are both alive and dead.

In this expository talk, I will explain quantum computing the way that I wish it had been explained to me. I will describe the basic operations and a compositional semantics for them. This is the basis for the so-called “structural” approaches to quantum theory, which have found applications in quantum programming languages, quantum error correction, and formal verification of quantum programs. I will assume no prior knowledge except for some linear algebra.

Abstracts of invited talks in the Special Session on Logic in Computer Science and Computability Theory

Abstracts of invited talks in the Special Session on Model Theory

▸SEBASTIAN ETEROVIĆ, A model-theoretic approach to the study of transcendental functions.

School of Mathematics, University of Leeds, Woodhouse Lane, Leeds, United Kingdom.

E-mail: s.eterovic@leeds.ac.uk.

Many important transcendental functions are used in number theory to study the behavior of certain sets of algebraic numbers. For example, the complex exponential function is used to study the roots of unity.

In this talk I will discuss a few model-theoretic approaches used to study transcendental functions coming from o-minimality, differential algebra, and abstract elementary classes. In each case, I will describe some of the contributions my collaborators and I have made in this area.

▸SCOTT MUTCHNIK, The existence axiom.

Department of Mathematics, Statistics and Computer Science, University of Illinois Chicago, USA.

E-mail: mutchnik@uic.edu.

One of the core projects of model theory asks: what can Shelah’s stability theory tell us about first-order theories that are not stable, or even simple? Kim-independence, introduced by Kaplan and Ramsey, extends stability-theoretic phenomena beyond the context of simple theories: namely to theories without the first/second strict order property/first tree property, or NSOP $_1$ theories. However, in contrast to simplicity theory, Kim-independence in general NSOP $_1$ theories has only been developed over models, rather than sets. Dobrowolski, Kim, and Ramsey, motivated by the definition of Kim-independence over models, offer one approach to defining Kim-independence over sets: they assume the existence of forking extensions over sets, the existence axiom, develop Kim-independence in that context, and ask whether all NSOP $_1$ theories satisfy the existence axiom. We explain the motivation for this question and discuss our answer, exhibiting an NSOP $_1$ theory violating the existence axiom, in fact the first known example of a theory without the strict order property violating this axiom.

Abstracts of invited talks in the Special Session on Nonclassical Logics

▸NOEMÍ LUBOMIRSKY $^*$ , PAULA MENCHÓN AND HERNÁN JAVIER SAN MARTÍN, Vakarelov’s construction for subresiduated lattices.

CONICET, Department of Mathematics, National University of La Plata, Provincia de Buenos Aires, Argentina.

E-mail: nlubomirsky@mate.unlp.edu.ar.

Department of Logic, Nicolaus Copernicus University in Toruń, Toruń, Poland.

CONICET, Department of Mathematics, National University of La Plata, Argentina.

Nelson’s constructive logic with strong negation, denoted by $\mathcal {N}$ is a well-known non-classical logic that combines the constructive approach of positive intuitionistic logic with a classical negation (see [6]). The algebraic models of $\mathcal {N}$ , form a variety called Nelson algebras or Nelson residuated lattices (see [7]). One of the main algebraic insights on this variety came with the realisation (independently due to M. Fidel and D. Vakarelov, see [10]) that every Nelson algebra can be represented as a special binary product (called a twist-structure) of a Heyting algebra. This correspondence was formulated as a categorical equivalence (see [9]) between Nelson algebras and a category of enriched Heyting algebras, which made it possible to transfer a number of fundamental results from the more widely studied theory of intuitionistic algebras to the realm of Nelson algebras.

Subresiduated lattices, which are a generalization of Heyting algebras, were introduced by G. Epstein and A. Horn [4] as an algebraic counterpart of some logics with strong implication previously studied by C. Lewy and I. Hacking [5]. These logics are examples of subuintuitionistic logics, i.e., logics in the language of intuitionistic logic that are defined semantically by using Kripke models, in the same way as intuitionistic logic is defined, but without requiring of the models some of the properties required in the intuitionistic case. Also in relation with the study of subintuitionistic logics, Celani and Jansana [2] got these algebras as the elements of a subvariety of weak Heyting algebras (see also [1, 3]). It is known that the variety $\mathsf {S4}$ , whose members are the S4-algebras, is the algebraic semantics of the modal logic $\mathbf {S4}$ . This means that $\phi $ is a theorem of $\mathbf {S4}$ if and only if the variety $\mathsf {S4}$ satisfies $\phi \approx 1$ . The variety of subresiduated lattices corresponds to the variety of algebras defined for all the equations $\phi \approx 1$ satisfied in the variety $\mathsf {S4}$ with the connectives conjunction $\wedge $ , disjunction $\vee $ , bottom $\perp $ , top $\top $ and a new connective of implication $\Rightarrow $ , called strict implication, defined by $\varphi \Rightarrow \psi := 1\rightarrow (\varphi \rightarrow \psi )$ , where $\rightarrow $ denotes the classical implication.

The main goal of the talk is to extend the Vakarelov’s construction, which establishes a link between Heyting algebras and Nelson algebras in the framework of subresiduated lattices. More precisely, we give a variety of algebras, whose members will be called subresiduated Nelson algebras, for which the original Vakarelov’s construction can be extrapolated by considering subresiduated lattices and subresiduated Nelson algebras, respectively.

[1] J. L. Castiglioni, V. Fernández, H. F. Mallea, and H. J. San Martín, On subreducts of subresiduated lattices and logic, Journal name spelled out, no abbreviations , arXiv:2211.02963 (2022).

[2] S. Celani and R. Jansana, Bounded distributive lattices with strict implication. Mathematical Logic Quarterly , vol. 51 (2005), no. 3, pp. 219–246.

[3] S. Celani, A. Nagy, and H. J. San Martín, Dualities for subresiduated lattices. Algebra Universalis , vol. 82 (2021), article 59.

[4] G. Epstein and A. Horn, Logics wich are characterized by subresiduated lattices. Zeitschirft für Mathematische Logik und Grundlagen der Mathematik , vol. 22 (1976), pp. 199–210.

[5] I. Hacking, What is strict implication? The Journal of Symbolic Logic , vol. 28 (1963), pp. 51–71.

[6] D. Nelson, Constructible falsity. The Journal of Symbolic Logic , vol. 14 (1949), pp. 16–26.

[7] H. Rasiowa, An algebraic approach to non-classical logics, Studies in Logic and the Foundations of Mathematics , vol. 78 , North-Holland, Amsterdam, 1974.

[8] A. Sendlewski, Some investigations of varieties of N-lattices. Studia Logica , vol. 43 (1984), pp. 257–280.

[9] A. Sendlewski, Nelson algebras through Heyting ones. I, Studia Logica , vol. 49 (1990), pp. 105–126.

[10] D. Vakarelov, Notes on N-lattices and constructive logic with strong negation. Studia Logica , vol. 36 (1977), pp. 109–125.

▸ELAINE PIMENTEL $^*$ , ROBERT FREIMAN, CARLOS OLARTE, AND CHRISTIAN FERMÜLLER, Reasoning about group polarization:from semantic games to sequent systems.

Department of Computer Science, University College London, London, UK.

E-mail: elaine.pimentel@gmail.com.

Group polarization—where the opinions or beliefs of individuals within a group become more extreme or polarized after interacting with each other—is rapidly gaining attraction, especially with the advent of social media platforms that have played a key role in the polarization of social, political, and democratic processes. This phenomenon is mainly studied in psychology [4, 2] and political philosophy [7, 8]. More recently, logicians have taken up the challenge of formal reasoning about social networks and changes in agents’s beliefs. Take for instance the Facebook logic [6] (an epistemic logic endowed with a symmetric “friendship” relation); the Tweeting logic [10] (formalizing announcements in a network); the logic for reasoning about social belief and change propagation [3], etc.

We focus on the modal logic PNL [9], which refers to Kripke frames with two types of disjoint and symmetric reachability relations. The individuals in a social network are identified with worlds of the frame, and they are related either as “friends” (positive) or as “enemies” (negative), but not both at the same time. These relationships can be understood in different ways: Instead of genuine friendship or enduring enmity, they may simply signify agreement or disagreement on a particular issue. Unlike the aforementioned logics, PNL was designed to reason about positive and negative relations among agents, a key aspect for defining and measuring polarization [1].

We take inspiration from [5], where PNL is extended in various ways to axiomatically characterize modally undefinable frame properties, including the disjointness of the two relations and collective connectedness.

Our approach to logical reasoning about group polarization is also based on PNL but focuses on a different aspect of formal reasoning about the corresponding models via games and proof systems. Games are a powerful tool to bridge the gap between intended and formal semantics, often offering a conceptually more natural approach to logic than the common paradigm of model-theoretic semantics.

We start by proposing a semantic game that characterizes the truth in a given network model. This provides an alternative to the standard definition of an evaluation function which supports a dynamic form of reasoning about concrete network models. We move on by arguing that effective formal reasoning with the relevant logics requires more than (just) Hilbert-style axiom systems. Rather, the automated search for proofs calls for Gentzen-style systems that respect (a restricted form of) the subformula property. Our method of transforming semantic games into proof games leads to the first Gentzen-style systems for variants of PNL, which modularly adapts to different frame properties by faithfully capturing the rules for elementary games.

[1] A. Bramson, P. Grim, D. J. Singer, W. J. Berger, G. Sack, S. Fisher, C. Flocken, and B. Holman, Understanding polarization: Meanings, measures, and model evaluation. Philosophy of Science , vol. 84 (2017), no. 1, pp. 115–159.

[2] D. J. Isenberg, Group polarization: A critical review and meta-analysis. Journal of Personality and Social Psychology , vol. 50 (1986), no. 6, pp. 1141–1151.

[3] F. Liu, J. Seligman, and P. Girard, Logical dynamics of belief change in the community. Synthese , vol. 191 (2014), no. 11, pp. 2403–2431.

[4] D. G. Myers and H. Lam, The group polarization phenomenon. Psychological Bulletin , vol. 83 (1976), no. 4, pp. 602–627.

[5] M. Y. Pedersen, S. Smets, and T. Ågotnes, Modal logics and group polarization. Journal of Logic and Computation , vol. 31 (2021), no. 8, pp. 2240–2269.

[6] J. Seligman, F. Liu, and P. Girard, Facebook and the epistemic logic of friendship, Theoretical Aspects of Rationality and Knowledge 2013 (Institute of Mathematical Sciences, Chennai, India), (Burkhard C. Schipper, editor), TARK, 2013, 10 pages.

[7] C. R. Sunstein, The law of group polarization, John M. Olin Program in Law and Economics Working Paper , no. 91 (1999), 39 pages.

[8] C. R. Sunstein, Group polarization and 12 angry men. Negotiation Journal , vol. 23 (2007), no. 4, pp. 53–75.

[9] Z. Xiong and T. Ågotnes, On the logic of balance in social networks. Journal of Logic, Language and Information , vol. 29 (2020), no. 1, pp. 53–75.

[10] Z. Xiong, T. Ågotnes, J. Seligman, and R. Zhu, Towards a logic of tweeting, Logic, Rationality and Interaction (Sapporo, Japan), (A. Baltag, J. Seligman, and T. Yamada, editors), Lecture Notes in Computer Science, 10455, Springer, 2017, pp. 49–64.

▸UMBERTO RIVIECCIO, (In)consistency operators in quasi-Nelson logic.

Departamento de Lógica, Historia y Filosofía de la Ciencia, UNED, Madrid, Spain.

E-mail: umberto@fsof.uned.es.

Logics of formal inconsistency (LFI) are among the most well-known and time-honoured inconsistency-tolerant, or paraconsistent, logical systems. Formally, an LFI is usually presented as a standard (propositional) consequence relation $(\vdash )$ over a language which includes a conjunction $(\land )$ , a disjunction $(\lor )$ , an implication $(\Rightarrow )$ , truth constants $(\bot , \top )$ and a negation $(\mathop {\sim })$ that crucially fails to satisfy the principle of explosion: $\varphi \land \mathop {\sim } \varphi \vdash \bot $ . To this language one usually adds a unary consistency connective $\circ $ that allows one to recover explosion in a more controlled way. The intended reading of $\circ \varphi $ is “ $\varphi $ is consistent”, and the following weaker principle is postulated:

(the finite gentle principle of explosion of [2, p. 50]) which can be informally read as follows: “if $\varphi $ is consistent and contradictory, then $\varphi $ explodes”.

In addition or (alternatively) to $\circ $ , a dual inconsistency connective $\bullet $ may be employed, and $\bullet \varphi $ is interpreted as “ $\varphi $ is inconsistent”. One may require $\bullet $ to be the negation-dual of $\circ $ (perhaps even taking $\bullet \varphi := \mathop {\sim } \circ \varphi $ as a definition) or may impose independent postulates on both. For instance, dualizing (1) one obtains:

which is indeed a valid principle of some LFI. Under some assumptions on the negation, it is also easily verified that (1) and (2) are equivalent via the definition $\bullet \varphi := \mathop {\sim } \circ \varphi $ (or its dual $\circ \varphi := \mathop {\sim } \bullet \varphi $ ).

The starting point of the present work is the paper [3], which investigates a family of LFI that result from adding a consistency connective to certain extensions of the Full Lambek Calculus with Exchange and Weakening ( $FL_{ew}$ ), the substructural logic determined by the class of commutative integral bounded residuated lattices. The approach of [3] is fairly general and modular, but it only applies to involutive logics (i.e., those hat satisfy the double negation law, $\mathop {\sim } \mathop {\sim } \varphi \Rightarrow \varphi $ ). The present work is a first attempt at extending this research project beyond the involutive setting.

To any given class $\mathsf {K}$ of residuated lattices one can associate two logical consequences in a standard way, the truth-preserving $\vdash ^{\top }_{\mathsf {K}}$ and the order-preserving $\vdash ^{\leq }_{\mathsf {K}}$ logic. Both share the same set of valid formulas, but in general they do not coincide: $\vdash ^{\top }_{\mathsf {K}}$ is the stronger and satisfies the principle of explosion, while $\vdash ^{\leq }_{\mathsf {K}}$ typically does not. The latter thus provides a natural candidate for an LFI based on residuated lattices. Assuming an axiomatization for $\vdash ^{\leq }_{\mathsf {K}}$ is given (see [11]), the paper [3] describes a method for axiomatizing the order-preserving logic $\vdash ^{\leq }_{\mathsf {K}^{\circ }}$ obtained by endowing each algebra in $\mathsf {K}$ with a consistency operator $\circ $ satisfying (the algebraic counterpart of) the gentle principle of explosion (1).

The procedure sketched above may be applied to several well-known order-preserving companions of substructural logics, including Łukasiewicz logic, Nelson’s constructive logic with strong negation [5] and nilpotent minimum logic [4]; in specific cases, further insight on the resulting logics is also gained thanks to the peculiar structure of prelinear algebras (for Łukasiewicz and nilpotent minimum) or the twist representation of Nelson algebras. The method of [3] does not apply, however, to many well-known logics based on residuated lattices—e.g., Hájek’s basic logic, product logic and $FL_{ew}$ itself—because the algebras in $\mathsf {K}$ are required to be distributive and involutive (see below for the relevant definitions). These limitations are due to technical reasons, and the main aim of the present contribution is indeed to explore the possibility of relaxing them.

We are going to show how the results of [3] can be extended if we take $\mathsf {K}$ to be the variety of quasi-Nelson algebras, a recently-introduced generalization of Nelson (and Heyting) algebras. Such a setting seems to be particularly promising for potential future research: for, on the one hand, quasi-Nelson algebras are distributive but not necessarily involutive residuated lattices; on the other, they can be represented through a Nelson-type twist construction that affords powerful insight into their structure. We therefore propose the present study as a preliminary investigation on the possibilities of extending the approach described above to more general classes of algebras and logics. We focus on laying the algebraic foundations, and in particular on the question of how to define and represent (in)consistency operators on quasi-Nelson algebras; we stress, however, that all the results we have established have a straightforward logical interpretation in the setting of logics extending $FL_{ew}$ .

[1] F. Bou, F. Esteva, J. M. Font, A. J. Gil, L. Godo, A. Torrens, and V. Verdú, Logics preserving degrees of truth from varieties of residuated lattices. Journal of Logic and Computation , vol. 19 (2009), no. 6, pp. 1031–1069.

[2] W. A. Carnielli and J. Marcos, A taxonomy of C-systems, Paraconsistency: The Logical Way to the Inconsistent (W. A. Carnielli, M. E. Coniglio, and I. M. Loffredo D’Ottaviano, editors), Lecture Notes in Pure and Applied Mathematics, vol. 228, CRC Press, Boca Raton, FL, 2002, pp. 1–94.

[3] F. Esteva, A. Figallo-Orellano, T. Flaminio, and L. Godo, Logics of formal inconsistency based on distributive involutive residuated lattices. Journal of Logic and Computation , vol. 31 (2021), no. 5, pp. 1226–1265.

[4] F. Esteva and L. Godo, Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy Sets and Systems , vol. 124 (2001), no. 3, pp. 271–288.

[5] D. Nelson, Constructible falsity. The Journal of Symbolic Logic , vol. 14 (1949), pp. 16–26.

Abstracts of invited talks in the Special Session on Philosophical Logic

▸NATALIA BUACAR, Normativity of logic: reasoning and argumentation.

Departamento de Filosofía, Universidad de Buenos Aires, Buenos Aires, Argentina.

E-mail: nbuacar@gmail.com.

In its most common characterization, logic is concerned with arguments. Specifically, it focuses on determining which arguments are valid or, more broadly, correct. This broad characterization introduces a normative aspect of logic but also raises questions about the object of logic. Arguments, complex linguistic entities composed of premises and a conclusion, are integral to various human activities, particularly reasoning and argumentation. The possible normative role of logic emerges in the context of these activities.

Reasoning and argumentation involve arguments but serve different goals and have distinct natures. According to a generalized internalist approach to reasoning, it occurs in the private realm and deals with beliefs, typically aiming to derive information from other information. In contrast, argumentation usually takes place in the public domain and has a dialogical character, where participants engage in the process of giving and asking for reasons ([1]) through utterances that may aim to persuade, negotiate, inquire, deliberate, or seek information ([13]).

The discussion surrounding the normativity of logic often focuses on explaining how logic is normative for reasoning ([6, 7, 4, 11]). Few works have focused on the normativity of logic in relation to argumentation ([8, 10, 3, 9]). Even if, prima facie, the public domain of communicative practices appears to be a more suitable context for finding a normative role for logic. Since it is not clear that we can always decide what to believe ([14]), it is challenging to norm what we ought to believe. These two ways of framing the problem may lead to different conclusions, as requirements that are reasonable in the public sphere may not carry the same weight in a private setting.

In this talk, I assume the distinction between two senses of normativity: evaluative and prescriptive ([12]). First, I revisit a general account of normativity that I have previously defended ([2]), in which logic is normative in an evaluative sense. Then, taking into account differences between reasoning and argumentation, I consider if an additional sense of normativity in prescriptive terms can arise from logic itself. I argue that it cannot and defend that this extra sense can come from general epistemic principles in reasoning and from cooperative principles ([5]) in argumentation.

[1] R. Brandom, Articulating Reasons: An Introduction to Inferentialism , Harvard University Press, Cambridge, 2000.

[2] N. Buacar, Lógica, justificación y normatividad. Análisis Filosófico , vol. 40 (2021), pp. 159–182.

[3] C. Dutilh Novaes, A dialogical, multi-agent account of the normativity of logic. Dialectica , vol. 69 (2015), no. 4, pp. 587–609.

[4] H.Field, What is the normative role of logic. Proceedings of the Aristotelian Society , vol. 83 (2009), pp. 251–268.

[5] H. P. Grice, Studies in the Way of Words , Harvard University Press, Cambridge, 1989.

[6] G. Harman, Logic and reasoning. Synthese , vol. 60 (1984), pp. 107–127.

[7] J. MacFarlane, In what sense (if any) Is logic normative for thought? unpublished manuscript.

[8] J. Mackenzie, Reasoning and logic. Synthese , vol. 79 (1989), pp. 99–117.

[9] A. Massolo, The normative role of logic for reasoning. THEORIA: An International Journal for Theory, History and Foundations of Science , vol. 38 (2023), no. 2, pp. 137–154.

[10] P. Milne, What is the normative role of logic. Proceedings of the Aristotelian Society , vol. 83 (2009), pp. 269–298.

[11] D. Tajer, The Normative Autonomy of Logic. Erkenntnis , vol. 87 (2022), no. 6, pp. 2661–2684.

[12] G. H. von Wright, Norm and Action: A Logical Enquiry , The Humanities Press, 1963.

[13] D. Walton, Informal Logic: A Handbook for Critical Argumentation , Cambridge University Press, Cambridge, 1989.

[14] B. Williams, Deciding to believe, Problems of the Self , Cambridge University Press, Cambridge, 1973, pp. 136–151.

▸JONAS R. BECKER ARENHART, What can stories teach us about logic?

Department of Philosophy, Federal University of Santa Catarina, Florianópolis, Brazil.

E-mail: jonas.becker2@gmail.com.

It is widely known that stories may be useful to teach us some important lessons, e.g., about morals or about society; more recently, it has become an interesting topic of discussion whether stories can be also useful in Logic. Given some assumptions common in the field of philosophy of logic, in particular that we should not confuse what follows from what with our theories about logical consequence, some authors have been suggesting that stories may teach us some important lessons concerning what actually follows from what. We are going to address this problem in this talk. We first briefly examine one position advanced by Andreas Kapsner in [2]. In short, Kapsner proposes that stories are a useful tool for the discovery of what follows from what. Some stories, furthermore, indicate that classical logic cannot be right, and Kapsner suggests candidates he considers more suitable for being the correct logic. In opposition to that view, Graham Priest [4, 5] has suggested that stories introduce an environment where one can indeed have some validities, but one should not confuse what is actually valid with what is valid in the story; one is not allowed to infer from stories to reality (despite some remarks that seem to point in that direction in [3]). Finally, we advance a view of one of the pioneers in the use of though experiments in logic, Newton da Costa, as discussed in [1]. We argue that da Costa has drawn conclusions from stories that undermine the basic background assumption causing trouble in the debate, the idea that there is something like validity outside of our theories. We then explore how to make sense methodologically of stories as kinds of thought experiments in logic.

[1] N. C. A. da Costa, Ensaio Sobre os Fundamentos da Lógica , São Paulo, Hucitec, 1980.

[2] A. Kapsner, The stories of logics. Australasian Journal of Logic , vol. 16 (2019), no. 4, pp. 133–158.

[3] G. Priest, Sylvan’s box: A short story and ten morals. Notre Dame Journal of Formal Logic , vol. 38 (1999), no. 4, pp. 573–582.

[4] G. Priest, Mathematical pluralism. Logic Journal of the IGPL , vol. 21 (2013), no. 1, pp. 4–13.

[5] G. Priest, A note on mathematical pluralism and logical pluralism. Synthese , vol. 198 (2021), supp. 20, pp. S4937–S4946.

▸JOÃO MARCOS, What is this thing called substructural logic?

Department of Philosophy, Federal University of Santa Catarina, Florianópolis, Brazil.

E-mail: botocudo@gmail.com.

The foundational approach championed by contemporary logic for constructing, precisely formalizing, and rigorously analyzing theories across various disciplines has been profoundly shaped by Tarski’s notion that logic itself can be axiomatized as a theory of consequence ([9]). A consequence relation, in this approach, may be canonically—yet always circumstantially!—presented by way of some deductive system or determined by some class of logical matrices. The attribute of being ‘formal’ is typically embodied, in practice, through a property of structurality that is realized by deductive systems that are schematic, in one way or another, or by formal semantics that are closed for substitutions ([8, 11]). Accordingly, ‘structural’ consequence relations are now increasingly known as ‘substitution-invariant’ consequence relations ([7]).

Meanwhile, in the context of the literature on Proof Theory, due to the capital influence of Gentzen ([6]), the term structural gained a very specific meaning ([10]), as a property of certain rules that could be formulated free from the presence of logical constants. In this tradition, ‘structural rules’ are simply opposed to ‘logical rules’. It is not entirely clear, of course, how a property of a certain proof system translates into a property of the consequence relation presented by the former system, and, indeed, numerous examples show that this is often simply not the case. In this context, thus, when proof systems in which certain structural rules are not present are under scrutiny, the term ‘substructural proof formalisms’ would clearly seem more to the point.

More recently, yet another usage for the term structurality gained traction, in a very prolific line of investigation that challenges the very definition of logical consequence, with robust applications to the debate concerning logical disputes, logical disagreement and logical pluralism, as well as to the study of paradoxes of vagueness and of truth. In particular, assuming that a notion of consequence, and the associated notion of logical validity, is regulated by the collection of ‘inferences’ that are sanctioned by it and by the ‘metainferences’ that encode, at a higher level, the deducibility steps to be allowed between the inferences at the lower level, a hierarchy of levels of consequence has been successfully investigated based on the control of ‘structurality’, in this latter sense ([4, 3, 1]).

A conceptual clarification of the points of contact between these three traditions, as well as of their points of departure, is imperative. In this talk I will bring terminological problems to the fore, discussing inadequacies that are prone to causing serious misunderstanding, and I will discuss a minimalistic many-dimensional framework that is capable of elegantly accommodating, from an abstract, a deductive and a semantical perspectives, consequence relations, such as the so-called ‘mixed consequence relations’ ([5]), that deviate from the Tarskian standard, as well as the logics that exemplify such alternative standards ([2]).

[1] E. Barrio and P. Egré, Editorial introduction: Substructural logics and metainferences. Journal of Philosophical Logic , vol. 51 (2022), no. 6, pp. 1215–1231.

[2] C. Blasio, J. Marcos, and H. Wansing, An inferentially many-valued two-dimensional notion of entailment. Bulletin of the Section of Logic , vol. 46 (2017), no. 3/4, pp. 233–272.

[3] E. A. Barrio, F. Pailos, and J. Calderón, Anti-exceptionalism, truth and the BA-plan. Synthese , vol. 199 (2021), no. 5, pp. 12561–12586.

[4] E. Barrio, L. Rosenblatt, and D. Tajer, The logics of strict-tolerant logic. Journal of Philosophical Logic , vol. 44 (2015), pp. 551–571.

[5] p. Cobreros, P. Egré, D. Ripley, and R. Van Rooij, Tolerant, classical, strict. Journal of Philosophical Logic , vol. 41 (2012), pp. 347–385.

[6] G. Gentzen, Untersuchungen über das logische Schließen. Mathematische Zeitschrift , vol. 35 (1935), pp. 176–210, 405–431.

[7] H. Lloyd, The Connectives , MIT Press, Cambridge, 2011.

[8] J. Łoś and R. Suszko, Remarks on sentential logics. Indagationes mathematicae , vol. 20 (1958), p. 5.

[9] J. Łukasiewicz and A. Tarski, Untersuchungen über den Aussagenkalkül. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III , vol. 23 (1930).

[10] G. Restall, Substructural Logics, (E. N. Zalta, editor), The Stanford Encyclopedia of Philosophy Metaphysics Research Lab, Stanford University, 2018.

[11] R. Wójcicki, Theory of Logical Calculi: Basic Theory of Consequence Operations , Synthese Library, 199, Springer, New York, 1988.

▸RICARDO O. RODRIGUEZ, Many-valued modal logics for reasoning with uncertain and incomplete knowledge.

UBA-FCEyN Computer Science Department and UBA-CONICET Computer Science Institute, Buenos Aires, Argentina.

E-mail: ricardo@dc.uba.ar.

Modal logics are well-known formalisms that account for representing and reasoning about notions such as necessity, belief, uncertainty, knowledge, similarity, obligations, time, etc. On the other hand, many-valued logical systems under the umbrella of the so-called mathematical fuzzy logic (in the sense of Hájek [6, 7]) appear as suitable logical frameworks to formalize reasoning with gradual properties, i.e., notions whose satisfaction is a matter of degree. Therefore, if one is interested in reasoning involving both gradualness and some sort of modalities one is led to study systems of many-valued modal logic. In this talk we will mainly survey various families of many-valued or fuzzy modal logic of different nature that have been proposed in the literature to reason about different models of uncertainty in the sense of graded belief [10, 11] and about preferences [14], both qualitatively and quantitatively.

In this presentation, we focus on families of many-valued modal logics following a more traditional approach, after Fitting’s pioneering works [8, 9], with a classical-like language extended with modal operators and with Kripke-style semantics, where possible worlds and the accessibility relation are many-valued, see e.g., [1, 4]. These systems are relatively simple to axiomatize and, in addition, they allow both nested modalities and mixed propositional and modal formulas.

Representative examples of these systems are Caicedo and Rodriguez’s minimal modal logics based on Gödel [0;1]-valued logic [4]. An alternative semantic for that logics is considered in [5] which allows to show the Finite Model Propert of them. In addition, a KD45-like extension has been recently shown to be adequate to reason about possibilistic uncertainty [12]. Another interesting examples are n-Lukasiewicz modal logics studied by Busaniche et al. in [2] and [3]. Finally, we will discuss different systems in this setting as well as some recent negative results by Vidal about non-axiomatizability of modal logics based on certain many-valued logics [13].

The author acknowledges partial support by the MOSAIC project (EU H2020-MSCA-RISE-2020 Project 101007627) and the PICT-2019-2019-00882 argentinean project.

[1] F. Bou, F. Esteva, L. Godo, and R. Rodriguez, On the minimum many-valued modal logic over a finite residuated lattice. Journal of Logic and Computation , vol. 21 (2011), no. 5, pp. 739–790.

[2] M. Busaniche, P. Cordero, and R.O. Rodriguez, Algebraic semantics for the minimum many-valued modal logic over Ln. International Journal On Fuzzy Sets and Systems , vol. 431 (2021), no. 8, pp. 94–109.

[3] M. Busaniche, P. Cordero, M. Marcos, and R.O. Rodriguez, An algebraic semantics for possibilistic finite-valued Lukasiewicz logic. International Journal Approximate Reasoning , vol. 159 (2023), article 108924.

[4] X. Caicedo and R. Rodriguez, Bi-modal Gödel modal logics. Journal of Logic and Computation , vol. 25 (2015), no. 1, pp. 37–55.

[5] X. Caicedo, G. Metcalfe, R. Rodriguez, and J. Rogger, A finite model property for Gödel modal logics, Logic, Language, Information, and Computation (WoLLIC 2013) (Address of meeting), (L. Libkin, U. Kohlenbach and R. de Queiroz, editors), Lecture Notes in Computer Science, 8071, Springer, New York, 2013, pp. 226–237.

[6] P. Cintula, P. Hájek, and C. Noguera (eds.), Handbook of Mathematical Fuzzy Logic, Volumes 1 and 2 , College Publications, London, 2011.

[7] P. Cintula, C. Fermüller, and C. Noguera (eds.), Handbook of Mathematical Fuzzy Logic, Volume 3 , College Publications, London, 2016.

[8] M. Fitting, Many-valued modal logics. Fundamenta Informaticae , vol. 15 (1991), pp. 325–354.

[9] M. Fitting, Many-valued modal logics II. Fundamenta Informaticae , vol. 17 (1992), pp. 55–73.

[10] L. Godo, P. Hájek, and F. Esteva, A fuzzy modal logic for belief functions. Fundamenta Informaticae , vol. 57 (2003), no. 2-4, pp. 127–146.

[11] P. Hájek, Metamathematics of Fuzzy Logic , Trends in Logic, 4, Kluwer, 1998.

[12] R. Rodriguez, O. Tuyt, F. Esteva, and L. Godo, Simplified Kripke semantics for K45-like Gödel modal logics and its axiomatic extensions. Studia Logica , vol. 110 (2022), no. 4, pp. 1081–1114.

[13] A. Vidal, Undecidability and non-axiomatizability of modal many-valued logics. The Journal of Symbolic Logic , vol. 87 (2022), no. 4, pp. 1576–1605.

[14] A. Vidal, F. Esteva, and L. Godo, Axiomatizing logics of fuzzy preferences using graded modalities. International Journal On Fuzzy Sets and Systems , vol. 401 (2020), pp. 163–188.

▸GISELE SECCO, Loparić avec les lacaniens: notes on a translation.

California State University San Bernardino.

E-mail: Dalva.Gisele@csusb.edu.

Andrea Loparić (1941–2021) was a logician who educated generations of Brazilian philosophers, contributed to the consolidation of the Brazilian logic community, and played a significant role in the early developments of non-classical logic in South America. Since her youth until the last year of her life, Andrea was passionately engaged in politically relevant dialogs and actions (having created a pioneering internet service by the end of the 1990’s) [5]. One of Loparić’s most important results in logic was a semantics for the minimal, or positive intuitionistic logic (PIL). Since Da Costa’s C $\omega $ -logic is an extension of PIL, determining a semantics for minimal logic was crucial. Using her previously developed theory of valuations, Loparić showed in [2] that systems admitting formal inconsistency without trivialization are correct and complete; applying her methods, she also proved the decidability of Da Costa and Arruda’s paraconsistent calculus. Among her last ongoing projects was a book celebrating her 80th birthday [4]—a selection of her logical papers, original contributions about her work plus an interview with her friend and colleague Ítala D’Ottaviano (“Dialogues on Logic, Philosophy, and Justice”). My contribution to this special session is a commentary on a rather extraordinary text in Loparić’s corpus that I am currently translating from French into Portuguese—and that does not appear in her festive volume. In [3], the result of her contribution to the conference Lacan avec les philosophes, Loparić shows that the famous and widely misinterpreted “formulae of sexuation” proposed by Jacques Lacan [1] can be given a strictly logical reading. By means of a semantics for non-classical predicate calculus in which identity and quantification are defined alternatively and the notions of interpretation and satisfiability are more generally described, Loparić makes Lacan’s formulae logically readable and perhaps psychoanalytically useful.

[1] J. Lacan and J.-A. Miller, Le séminaire de Jacques Lacan, Livre XX Encore: 1972–1973 , Éditions du Seuil, Paris, 1975.

[2] A. M. A. C. Loparić, A semantical study of some propositional calculi. The Journal of Non-Classical Logic , vol. 3 (1986), no. 1, pp. 74–95.

[3] A. Loparić, Les négations et les univers de discours, Lacan avec les Philosophes , Éditions du Albin Michel, Paris, 1991.

[4] A. Rodrigues, C. Mortari, E. Pimentel, and G. D. Secco, Andrea Lopari? on semantics for non-classical logics, Unicamp: Coleção, forthcoming.

[5] G. D. Secco, Rede Brasileira de Mulheres Filósofas Andrea Loparić, Filósofas Brasileiras, youtube video.

Abstracts of invited talks in the Special Session on Set Theory

▸MIGUEL A. CARDONA, On the constant evasion and prediction numbers.

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel.

E-mail: miguel.cardona@mail.huji.ac.il.

Call a function $\pi \colon {}^{<\omega }2\to 2$ a predictor. We denote by $\Pi _2$ the set of predictors $\pi \colon {}^{<\omega }2\to 2$ . Given $\pi \in \Pi _2$ and $f\in {}^{\omega }2$ , say that $\pi $ predicts constantly f denoted by $f\sqsubset ^{\mathrm {pc}} \pi $ iff $\exists k\in \omega \,\forall ^\infty i\,\exists j\in [i,i+k)\colon f(j)=\pi (f{\upharpoonright }j)$ .

We define the constant evasion number

and the constant prediction number

These cardinals have been intensively researched by many people who have studied for decades the relation between these cardinal invariants and other classical cardinal invariants of the continuum. In particular, it is known in ZFC that $\mathfrak{e}_2^{\mathrm {const}}\leq \mathrm {non}(\mathcal {E})$ and $\mathrm {cov}(\mathcal {E})\leq \mathfrak{v}_2^{\mathrm {const}}$ due to Kamo [6], where $\mathrm {non}(\mathcal {E})$ and $\mathrm {cov}(\mathcal {E})$ denotes the uniformity number and the covering number of the ideal $\mathcal {E}$ generated by the $F_\sigma $ measure zero subsets of the reals.

On the other hand, regarding the consistency Brendle [1] proved $\mathrm {CON}(\mathfrak{e}_2^{\mathrm {const}}>\mathfrak{b})$ where $\mathfrak{b}$ is the bounding number; Kamo [6, 7] prove $\mathrm {CON}(\mathfrak{d}<\mathfrak{v}_2^{\mathrm {const}})$ and $\mathfrak{v}_2^{\mathrm {const}}>\mathrm {cof}(\mathcal {N})$ where $\mathfrak{d}$ and $\mathrm {cof}(\mathcal {N})$ denotes the dominating number and the cofinality of $\mathcal {N}$ the null ideal, respectively; and Brendle and Shelah [4] proved $\mathrm {CON}(\mathfrak{e}_2^{\mathrm {const}}<\mathrm {add}(\mathcal {N}))$ where $\mathrm {add}(\mathcal {N})$ is the additivity of $\mathcal {N}$ the null ideal.

In addition, concerning the consistency of the constant evasion and prediction numbers, we prove $\aleph _1 < \mathrm {add}(\mathcal {N}) < \mathfrak{e}_2^{\mathrm {const}} < \mathfrak{b} < 2^{\aleph _0}$ . This work is joint with Miroslav Repický and Saharon Shelah [5].

[1] J. Brendle, Evasion and prediction—The Specker phenomenon and Gross space. Forum mathematicum , vol. 7 (1995), no. 5, pp. 513–542.

[2] J. Brendle, Evasion and prediction. III. Constant prediction and dominating reals. Journal of the Mathematical Society of Japan , vol. 55 (2003), no. 1, pp. 101–115.

[3] J. Brendle and S. Shelah, Evasion and prediction II. Journal of the London Mathematical Society , vol. 53 (1996), no. X1, pp. 19–27.

[4] J. Brendle and S. Shelah, Evasion and prediction. IV. Strong forms of constant prediction. Archive for Mathematical Logic , vol. 42 (2003), no. 4, pp. 349–360.

[5] M. A. Cardona, S. Shelah, and R. Miroslav, Separating the left side of Cichoń’s diagram along with $\mathfrak{e}_2^{\mathrm {const}}$ , in preparation.

[6] S. Kamo, Cardinal invariants associated with predictors, Logic Colloquium ‘98 (Prague), (S. Buss, P. Hájek, and P. Pudlák, editors), Lecture Notes in Logic, vol. 13, AK Peters, 2000, pp. 280–295.

[7] S. Kamo, Cardinal invariants associated with predictors. II. Journal of the Mathematical Society of Japan , vol. 53 (2001), no. 1, pp. 35–57.

▸ALEJANDRO POVEDA, Axiom $\mathcal {A}$ and supercompactness.

Department of Mathematics and Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, USA.

E-mail: alejandro@cmsa.fas.harvard.edu.

A cardinal $\delta $ is called $C^{(1)}$ -supercompact if for each cardinal $\lambda>\delta $ there is an elementary embedding $j\colon V\rightarrow M$ with $\text {crit}(j)=\delta $ , $ j(\delta )>\lambda $ , $M^\lambda \subseteq M$ and $j(\delta )$ strong limit (Bagaria, 2012). In this talk I will discuss the relationship between $C^{(1)}$ -supercompactness and classical supercompactness. Our main result is the consistency of every supercompact cardinal being $C^{(1)}$ -supercompact. This is a consequence of a new axiom we introduce called $\mathcal {A}$ (P. and Woodin, 2023). We show that this axiom is consistent with virtually all known large cardinals. Axiom $\mathcal {A}$ is tightly connected to the following inner-model problem: Suppose that $\delta $ is a cardinal carrying an elementary embedding $j\colon V\rightarrow M$ with $\text {crit}(j)=\delta $ , $M^\delta \subseteq M$ and $j(\delta )$ strong limit — must $\delta $ be superstrong with target $j(\delta )$ ? Woodin has conjectured (2018) that under $" V=\mathrm {Ultimate}$ -L” the answer is affirmative. As a result, Woodin’s conjecture implies that, under $" V=\mathrm {Ultimate}$ -L”, every $C^{(1)}$ -supercompact cardinal is a limit of supercompacts. Should this conjecture be true, axiom $\mathcal {A}$ (while compatible with all known large cardinals) would yield a set-theoretic universe very far from $"\mathrm {Ultimate}$ -L”. If time permits, we shall also discuss a strengthening of axiom $\mathcal {A}$ .

Abstracts of contributed talks

▸DIEGO BEJARANO, A robust(er) Scott rank for separable metric structures.

Group in Logic and the Methodology of Science, University of California Berkeley, Berkeley, CA, USA.

E-mail: bejarano@berkeley.edu.

In [1], Ben Yaacov et. al. extended the basic ideas of Scott analysis to metric structures in infinitary continuous logic. These include back-and-forth relations, Scott sentences, and the Lopez–Escobar theorem to name a few.

In this talk, I will talk my work connecting the ideas of Scott analysis to the definability of automorphism orbits and a notion of isolation for types within separable metric structures.

Our results are a continuous analog of the robuster Scott rank developed by Montalbán in [2] for countable structures in discrete infinitary logic. However, there are some differences arising from the subtleties behind the notion of definability in continuous logic.

[1] I. Ben Yaacov, M. Doucha, A. Nies, and T. Tsankov, Metric Scott analysis. Advances in Mathematics , vol. 318 (2017), no. X, pp. 46–87.

[2] A. Montalbá, A robuster Scott rank. Proceedings of the American Mathematical Society , vol. 143 (2015), no. 12, pp. 5427–5436.

▸JULIÁN C. CANO, Some combinatorial properties of semiselective ideals.

Departamento de Matemáticas, Universidad de los Andres, Bogotá, Colombia.

E-mail: jc.canor@uniandes.edu.co.

Local Ramsey Theory, developed mainly by Mathias, Matet, Farah, and Todorčević (see [3, 4, 5, 7, 8]), studies the combinatorial properties of two types of families of infinite subsets of integers: selective and semiselective ideals. Its applications and interactions with ‘classical’ Ramsey theory (see [2, 6]) allow to explore and construct a localized version of Ellentuck space, where homogeneity of Borel partitions of $[\omega ]^{\omega }$ is delimited and localized in a fixed coideal; in this way, it is possible to give a topological characterization of the local Ramsey property through an abstract Baire property associated with ideals (see [3, 8]).

In this talk, we will present a general overview of the combinatorial structure of local Ramsey spaces, where we will propose an alternative proof of the semiselective Ellentuck theorem without using local combinatorial forcing. Also, we will give some results about additivity and distributivity related to the $\sigma $ -ideal of all Ramsey null sets as well as the generalized Suslin operation on the $\sigma $ -algebra of all Ramsey sets with respect to semiselective ideals. Finally, we prove as the main result that under CH and $\neg $ SH, there is a semiselective coideal that does not contain any selective ultrafilter, although it is consistent that every semiselective coideal contains a selective ultrafilter; so, it should be noted that our results answers a questions from Farah and Todorčević.

This is joint work with Carlos Di Prisco and Michael Hrušák (see [1]).

[1] J. C. Cano, C. A. di Prisco, and M. Hrušák, Some combinatorial properties of semiselective ideals, submitted.

[2] E. Ellentuck, A new proof that analytic sets are Ramsey. The Journal of Symbolic Logic , vol. 39 (1974), no. 1, pp. 163–165.

[3] I. Farah, Semiselective coideals. Mathematika , vol. 45 (1998), no. 1, pp. 79–103.

[4] P. Matet, Happy families and completely Ramsey sets. Archive for Mathematical Logic , vol. 32 (1993), pp. 151–171.

[5] A. R. D. Mathias, Happy families. Annals of Mathematical Logic , vol. 12 (1977), no. 1, pp. 59–111.

[6] F. P. Ramsey, On a problem of formal logic. Proceedings of the London Mathematical Society , vol. 30 (1930), no. 4, pp. 264–286.

[7] S. Todorčević, Topics in Topology , Springer-Verlag, 1997.

[8] S. Todorčević, Introduction to Ramsey Spaces , Princeton University Press, Princeton, 2010.

▸EDWIN CELIS, Exploring universal compression in mathematical theories: insights from first-order logic and beyond.

Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Columbia.

E-mail: ercelism@unal.edu.co.

A desirable property in mathematical theories lies in the ability to possess minimal yet sufficiently relevant information to adapt to any expressive context of the theory. In the realm of first-order theories, we encounter UDTFS (uniform definability of types over finite sets) [5]. Laskowski has proved that $o$ –minimal theories exhibit this property, while Guingona [4] has established that $dp$ –minimal theories also possess it. On the other hand, Chernikov and Simon in [2] have shown that NIP theories also satisfy UDTFS. In this talk we first present with an analysis of these results and their relationship with VC-dimension, a concept introduced in [6]. Subsequently, we address in connection with the main topics of this lecture. The latest finding by Kaplan and Eshel [3] where they establish that a formula $\varphi (x;y)$ is NIP if and only if it exhibits the UDTFS property. This theoretical framework allows us to approach the initial question from different logical perspectives. To conclude, we will review the progress achieved in the field of continuous logics. [1]

[1] A. Anderson, Fuzzy VC Combinatorics and Distality in Continuous Logic, 2023, arXiv:2310.04393.

[2] A. Chernikov and P. Simon, Externally definable sets and dependent pairs II. Transactions of the American Mathematical Society , vol. 367 (2015), no. 7, pp. 5217–5235.

[3] S. Eshel and I. Kaplan, On uniform definability of types over finite sets for NIP formulas, 2020, arXiv:1904.10336.

[4] V. Guingona, On uniform definability of types over finite sets. The Journal of Symbolic Logic , vol. 77 (2012), no. 2, pp. 499–514.

[5] H. R. Johnson and M. C. Laskowski, Compression schemes, stable definable families, and o-minimal structures. Discrete & Computational Geometry , vol. 43 (2010), pp. 914–926.

[6] V. Vapnik, On the uniform convergence of relative frequencies of events to their probabilities. Doklady Akademii Nauk USSR , vol. 181 (1968), pp. 781–787.

▸NICOLÁS CUERVO OVALLE, Randomizations of $\omega $ -stable non-multidimensional theories and the SB-Property.

Departamento de Matemáticas, Universidad de Los Andes, Bogotá, Columbia.

E-mail: n.cuervo10@uniandes.edu.co.

We say that a complete theory T has the Schröder–Bernstein property, or simply, the SB-property, if for any two models $\mathcal {M},\mathcal {N}\models T$ that are elementarily bi-embeddable, i.e., there exist elementary embeddings $\varphi \colon \mathcal {M}\to \mathcal {N}$ and $\psi \colon \mathcal {N}\to \mathcal {M}$ , then $\mathcal {M}$ and $\mathcal {N}$ are isomorphic.

One motivation for studying the SB-property is that if T has this property, then T would be a theory for which we have a “good understanding” of its models, in terms that they are classified by some reasonable collection of invariants. The SB-property for first order discrete theories have been well studied in [4], [5], and [7]. For example, in [7] it was shown that, for discrete first-order theories,

The purpose of this talk is to study the Schröder–Bernstein property in the continuous context for Randomizations. Informally, a randomization of a first-order (discrete) model, $\mathcal {M}$ , is a two-sorted metric structure which consists of a sort of events and a sort of functions with values on the model, usually understood as random variables. More generally, given a complete first-order theory T, there is a complete continuous theory known as randomized theory, $T^R$ , which is the common theory of all randomizations of models of T. Randomizations were introduced first by Keisler in [6], and then axiomatized in the continuous setting by Ben Yaacov and Keisler in [3]. Since randomizations were introduced, many authors focused on examining which model theoretic properties of T are preserved on $T^R$ . For example, in [3] and [2], it was shown that properties like $\omega $ -categoricity, stability, and dependence are preserved. Similarly, in [1], it is proved that the existence of prime models is preserved by randomization, however notions like minimal models are not preserved. Following these ideas, we will provide some positive results and examples for which the SB-property is preserved under randomizations. Furthermore, we will prove that the non-multidimensionality of an $\omega $ -stable theory is preserved by randomizations, in the direction that the above theorem be also true for the continuous context.

[1] U. Andrews and H. Jerome Keisler, Separable models of randomizations. The Journal of Symbolic Logic , vol. 80 (2015), no. 4, pp. 1149–1181.

[2] I. Ben Yaacov, Continuous and random Vapnik-Chervonenkis classes. Israel Journal of Mathematics , vol. 173 (2009), no. 1, pp. 309–333.

[3] I. Ben Yaacov and H. J. Keisler, Randomizations of models as metric structures. Confluentes Mathematici , vol. 1 (2009), no. 2, pp. 197–223.

[4] J. Goodrick, When does elementary bi-embeddability imply isomorphism?, arXiv:0705.1849, 2007.

[5] J. Goodrick and M. Laskowski, The Schröder-Bernstein property for a-saturated models. Proceedings of the American Mathematical Society , vol. 142 (2014), no. 3, pp. 1013–1023.

[6] H. J. Keisler, Randomizing a model. Advances in Mathematics , vol. 143 (1999), no. 1, pp. 124–158.

[7] T. A. Nurmagambetov, Characterization of $\omega $ -stable theories of bounded dimension. Algebra and Logic , vol. 28 (1989), no. 5, pp. 388–396.

▸BRUNO DA RÉ $^*$ AND DAMIAN SZMUC, Non-deterministic matrices for semi-cocanonical deduction systems.

Department of Philosophy, University of Buenos Aires, and IIF-SADAF-CONICET, Argentina.

E-mail: bruno.horacio.da.re@gmail.com.

This article aims to dualize several results that Lahav has proved concerning various types (including possibly Cut-free and Identity-free systems) of canonical sequent calculi, systems equipped with well-behaved forms of left and right introduction rules for logical expressions. Instead, we focus on such kinds but rather of cocanonical sequent calculi, systems equipped with well-behaved forms of left and right elimination rules for logical expressions—which, simply put, proceed from sequents featuring complex formulae to their component subformulas. Usual canonical rules are the following:

Cocanonical rules involving usual vocabulary are, for instance, the following:

We will also consider some controlled versions of Cut and Identity, for instance:

Our main goals, are to prove soundness and completeness results for the target systems’ consequence relation in terms of their characteristic 3- or 4-valued non-deterministic matrices. These results will, incidentally, affect the potential repercussion of adding structural rules such as Cut or Identity to these calculi—phenomena usually described as Cut-admissibility and Identity-antiadmissibility, respectively.

▸FELIPE ESTRADA PELÁEZ. Model theory of homogeneous D-sets.

Departamento de Matemáticas, Universidad de Los Andes, Bogotá, Colombia.

E-mail: f.estrada@uniandes.edu.co.

A D-set is a model-theoretic structure which consists of a set, generally thought of as the leaves of a tree, and a quaternary relation D, such that the relation $D(a,b;c,d)$ holds if and only if the unique path from the leaf a to the leaf b does not intersect the path from leaf c to leaf d. D-sets arise naturally during the study of the automorphism groups of $\omega $ -categorical structures [1, 2].

D-sets have been studied extensively by Adeleke and Neumann [3], who provide a first-order axiomatization of their theory, but the model-theoretic properties of D-sets themselves have remained largely unexplored. During this talk I will present original results regarding these properties. In particular, as a main result I provide sufficient and necessary conditions for a D-set augmented with unary predicates to be ultrahomogeneous, as well as weaker notions of homogeneity, among other model-theoretic properties. These results are part of an in-development article on the model theory of D-sets.

[1] S. A. Adeleke and D. Macpherson, Classification of infinite primitive Jordan permutation group. Proceedings of the London Mathematical Society , vol. 3 (1996), no. 1, pp. 63–123.

[2] S. A. Adeleke and P. M. Neumann, Primitive permutation groups with primitive Jordan sets. Journal of the London Mathematical Society , vol. 53 (1996), no. 2, pp. 209–229.

[3] S. A. Adeleke and P. M. Neumann, Relations related to betweenness: Their structure and automorphisms. Memoirs of the American Mathematical Society , vol. 623 (1998), 125 pages.

▸CAMILLO FIORE $^*$ AND BRUNO DA RÉ, Maximally substructural classical logic.

Institute for Research in Philosophy, University of Buenos Aires and CONICET, Argentina.

E-mail: camillo.g.fiore@gmail.com.

Roughly speaking, structural principles are principles whose formulation does not require any constant of the object language. Classical logic validates a number of well-known structural principles, such as for instance

Here, $A,B,...$ are formulas of the relevant object language, $\Gamma ,\Delta ,...$ are multisets of formulas, and represents logical consequence. ‘C’ stands for ‘Contraction’, ‘W’ for ‘Weakening’ and ‘Id’ for ‘Identity’.

In the last years, we have seen the emergence of various logical systems that are coextensive with classical logic but in a certain sense invalidate some of the above structural principles. In the model-theoretic framework, these systems are obtained by finding a consequence relation that validates all and only the arguments that are valid in classical logic, but relative to which some of the structural principles are ‘locally invalid’, that is, fail to preserve satisfaction at every interpretation. Thus, for instance, Cobreros et. al. [2, 1] and Ripley [4, 5] present system $\mathbf {ST}$ , where Cut is locally invalid. Also, Rosenblatt [6] presents system $\mathbf {NC}$ , where not only Cut but also the principles of Contraction are locally invalid. Curiously, no system of this kind has been proposed were the principles of Weakening are locally invalid. Such a system seems possible in principle, since we know that there are sequent calculi for classical logic where the rules of Weakening are admissible but not derivable.

In this paper, we define a system whose consequence relation is coextensive with classical logic, but in which the principles of Cut, Contraction and Weakening are all locally invalid. We call this system $\mathbf {msCL}$ (for ‘Maximally Substructural Classical Logic’), and obtain it by means of a six-valued semantics and a sui generis definition of logical consequence. We show that $\mathbf {msCL}$ can be used to provide a semantics for the sequent calculus $\mathbf {G3}$ for classical logic (see Negri and von Plato [3]). Lastly, we show that the trick that we use to define $\mathbf {msCL}$ can be transposed to any other Tarskian Logic. That is, for any Tarskian logic $\mathbf {L}$ it is possible to define a logic $\mathbf {msL}$ which is coextensive to $\mathbf {L}$ but locally invalidates all structural principles except for Id.

[1] P. Cobreros, P. Égré, D. Ripley, and R. Van Rooij, Tolerance and mixed consequence in the S’valuationist setting. Studia Logica , vol. 100 (2012), no. 4, pp. 855–877.

[2] P. Cobreros, P. Égré, D. Ripley, and R. Van Rooij, Reaching transparent truth. Mind , vol. 122 (2013), no. 488, pp. 841–866.

[3] S. Negri and J. Von Plato, Structural Proof Theory , Cambridge University Press, Cambridge, 2008.

[4] D. Ripley, Conservatively extending classical logic with transparent truth. The Review of Symbolic Logic , vol. 5 (2012), no. 2, pp. 354–378.

[5] D. Ripley, Paradoxes and failures of cut. Australasian Journal of Philosophy , vol. 91 (2013), no. 1, pp. 139–164.

[6] L. Rosenblatt, Noncontractive classical logic. Notre Dame Journal of Formal Logic , vol. 60 (2019), no. 4, pp. 559–585.

▸RUSSELL MILLER, Computability in Galois groups of infinite algebraic field extensions.

Queens College & CUNY Graduate Center, New York, NY, USA.

E-mail: russell.miller@qc.cuny.edu.

When $F\subset E$ is an infinite algebraic field extension, it is natural to present the Galois group $\operatorname {Gal}(E/F)$ as the set of all paths through a tree. When E and F are computable fields (and the embedding $F\hookrightarrow E$ is computable), this tree is also computable and serves as an effective presentation of $\operatorname {Gal}(E/F)$ , even when this Galois group is uncountable.

Each automorphism $f\in \operatorname {Gal}(E/F)$ has a Turing degree, and certain of them will be computable. For each Turing ideal $\mathcal I$ , the set of automorphisms with degrees in $\mathcal I$ forms a subgroup of $\operatorname {Gal}(E/F)$ . We investigate the extent to which this may be an elementary subgroup, using certain specific examples. With the ground field $\mathbb Q$ , one naturally asks about both the case $E=\overline {\mathbb Q}$ (so $\operatorname {Gal}(E/F)$ is the absolute Galois group of $\mathbb Q$ , which is a difficult object to study) and the case where E is the cyclotomic field inside $\overline {\mathbb Q}$ , which is much better understood. Alternatively, with the ground field $F=K(t)$ , where K is algebraically closed and is extended by a transcendental t, natural questions about computability for the absolute Galois group of F often turn into purely group-theoretic questions. In each case, one may consider the subgroup defined by the minimal Turing ideal containing only the degree $\boldsymbol 0$ , or by other principal Turing ideals, or by a Scott ideal $\mathcal I$ (containing, for each $\boldsymbol d\in \mathcal I$ , some degree that is PA relative to $\boldsymbol d$ ).

▸NICOLÁS NÁJAR SALINAS, Abstract elementary classes and their logic.

Departamento de Matemáticas, Universidad Nacional de Colombia, Columbia.

E-mail: nicolas.na7ar@gmail.com.

The concept of Abstract Elementary Class was introduced by Shelah in the 1980’s ([4], [3]) as a semantical context in which we can develop the machinery to prove a Morley-like theorem to sentences of the logic $\mathbb {L}_{\omega _1,\omega }$ and much wider context with an adequate substructure relation. As we can see in [1], [2], [4] or [5], there are different ways to provide abstract elementary classes with a syntax, and use it to derive some structural results about AECs. Using some ideas of Kivimäi, Väänänen and Villaveces, we use the axiomatization given by Shelah and Villaveces in [5] to present an adequate logic for abstract elementary classes in which we can define order, independence and strict order like properties in a syntactical way.

[1] J. T. Baldwin and W. Boney, Hanf numbers and presentation theorems in AECs, Beyond First Order Model Theory (José Iovino, editor), CRC Press, Boca Raton, FL, 2017, pp. 327–352.

[2] Leung, Samson, Axiomatizing AECs and applications. Annals of Pure and Applied Logic , vol. 174 (2023), no. 5, paper 103248, 18 pages.

[3] S. Shelah, Universal classes, Classification Theory (Chicago, IL, December 1985) , (J.T. Baldwin, editor), Lecture Notes in Mathematics, 1292, Springer-Verlag, 1987, pp. 264–418.

[4] S. Shelah, Classification of nonelementary classes, II, Classification Theory (Chicago, IL, December 1985) , (J.T. Baldwin, editor), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, 1987, pp. 419–497.

[5] S. Shelah and A. Villaveces, Infinitary logics and abstract elementary classes. Proceedings of the American Mathematical Society , vol. 150 (2022), no. 1, pp. 371–380.

▸EUDES NAZIAZENO $^*$ , MARCO BARONE, AND NICOLÁS CARO-MONTOYA, Indecomposable polynomial rings interpreting arithmetic.

Universidade Federal de Pernambuco, Recife, Brazil.

E-mail: eudes.naziazeno@ufpe.br.

By using algebraic properties of (commutative unital) indecomposable polynomial rings we achieve results concerning their first-order theory, namely: interpretability of arithmetic in the language of rings without parameters. This extends the scope of a method due to Raphael Robinson, which deals with a restricted class of polynomial integral domains.

Some results in a similar setting (namely, rings containing transcendental-like elements) can be found in [4], [5], [14], [2], [11], [3], [13], [15], [9], [10], [8]. One may check that all these results are restricted to integral domains (in any characteristic); a few exceptions can be found on [1, 7, 12], where the rings considered are certain rings of fractions of finitely generated algebras over the rational integers, and on [6], where the authors consider certain rings of continuous functions.

We concentrate this talk on constructing a uniform two-dimensional interpretation of arithmetic in reduced indecomposable polynomial rings over a nonfield, and we briefly mention the corresponding interpretability results for the other types of reduced indecomposable polynomial rings. We end the construction removing the reducedness hypothesis by exploiting the fact that the nilradical $N_{S}$ of a polynomial ring S is uniformly definable and that if, in addition, S is indecomposable, then the corresponding factor ring $S / N_{S}$ is isomorphic to a reduced indecomposable polynomial ring.

[1] M. Aschenbrenner, A. Khélif, E. Naziazeno, and T. Scanlon, The logical complexity of finitely generated commutative rings. International Mathematics Research Notices , vol. 2020 (2018), no. 1, pp. 112–166.

[2] J. Becker, J. Denef, and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, Model Theory of Algebra and Arithmetic (Karpacz, 1979) , (L. Pacholski, J. Wierzejewski, and A. J. Wilkie, editors), Lecture Notes in Mathematics, 834, Springer, 1980, pp. 1–9.

[3] J. Demeyer, Recursively enumerable sets of polynomials over a finite field are Diophantine. Inventiones Mathematicae , vol. 170 (2007), no. 3, pp. 655–670.

[4] J. Denef, The Diophantine problem for polynomial rings and fields of rational functions. Transactions of the American Mathematical Society , vol. 242 (1978), pp. 391–399.

[5] J. Denef, The Diophantine problem for polynomial rings of positive characteristic, Logic Colloquium ‘78 (Mons, 1978) , (M. Boffa, D. van Dalen, and K. McAloon, editors), Studies in Logic and the Foundations of Mathematics, 97, North Holland, 1979, pp. 131–145.

[6] L. Darnière and M. Tressl, Defining integer-valued functions in rings of continuous definable functions over a topological field. Journal of Mathematical Logic , vol. 20 (2020), no. 3, paper 2050014, 24 pages.

[7] Eisenträger, Kirsten, Hilbert’s Tenth Problem and Arithmetic Geometry, Dissertation, University of California Berkeley, 2003.

[8] K. Eisenträger, Hilbert’s tenth problem for function fields of varieties over $\mathbb {C}$ . International Mathematics Research Notices , vol. 59 (2004), pp. 3191–3205.

[9] K. Eisenträger and A. Shlapentokh, Hilbert’s tenth problem over function fields of positive characteristic not containing the algebraic closure of a finite field. Journal of the European Mathematical Society , vol. 19 (2017), no. 7, pp. 2103–2138.

[10] K. H. Kim and F. W. Roush, Diophantine undecidability of $\mathbf {C}( t_1 , t_2 )$ . Journal of Algebra , vol. 150 (1992), no. 1, pp. 35–44.

[11] R. Miller and A. Shlapentokh, On existential definitions of c.e. subsets of rings of functions of characteristic 0. Annals of Pure and Applied Logic , vol. 173 (2022), no. 4, paper 103076, 50 pages.

[12] G. A. Noskov, Elementary theory of a finitely generated commutative ring. Mathematical Notes , (1983), pp. 12–15.

[13] Pasten, Hector, Pheidas, Thanases, and Vidaux, Xavier, Uniform existential interpretation of arithmetic in rings of functions of positive characteristic. Inventiones Mathematicae , vol. 196 (2014), no. 2, pp. 453–484.

[14] T. Pheidas, An undecidability result for power series rings of positive characteristic. II. Proceedings of the American Mathematical Society , vol. 100 (1987), no. 3, pp. 526–530.

[15] A. Shlapentokh, On Diophantine definability and decidability in some rings of algebraic functions of characteristic 0. The Journal of Symbolic Logic , vol. 67 (2002), no. 2, pp. 759–786.

▸JUAN RICARDO PRADA, On some properties of the relevant logic $E^{*}$ .

Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Universidad Nacional de la Plata, Provincia de Buenos Aires, Argentina.

E-mail: jrprada@ut.edu.co.

The relevant logic E was proposed by Anderson in [1] and Anderson–Belnap in [2] (Section 21.1).

Anderson in [1], and Anderson-Belnap in ([2], Section 23.5) show that E is equivalent to a Fitch-style proof system. On the other hand, in [6] and [7], another axiomatic system (which is assumed to be equivalent to logic E) is presented. We call this system $E^{*}$ . In this communication, we will show that the assumption made in [7] about the equivalence of the systems E and $E^{*}$ is not correct. The system $E^{*}$ is weaker than E. Furthermore, using the tools developed in [3], we propose a Fitch-style proof system for the logic $E^{*}$ . We will also analyse some properties of $E^{*}-$ matrices, as presented in [6] and [7].

Finally, we will show that the logic $E^{*}$ possesses the variable sharing property (VSP), which is a desirable characteristic of a relevant logic.

[1] A. R. Anderson, Completeness theorems for the systems E of entailment and EQ of entailment with quantification. Mathematical Logic Quarterly , vol. 6(1960), pp. 201–216.

[2] A. R. Anderson and N. D. Belnap, Entailment: The Logic of Relevance and Neccessity, Volume 1 , Princeton University Press, Princeton, NJ, 1975.

[3] R. T. Brady, Natural deduction systems for some quantified relevant logics. Logique et Analyse , vol. 27 (1984), no. 8, pp. 355–377.

[4] R. T. Brady (ed.) Relevant Logics and Their Rivals, Volume 2 , Aldershot, Ashgate, 2003.

[5] M. Tokarz, Essays in Matrix Semantics of Relevant Logics , The Polish Academy of Sciences, Poland, 1980.

[6] L. Typańska-Czajka, Two infinite sequences of pre-maximal extensions of the relevant logic E. Bulletin of the Section of Logic , vol. 48 (2019), no. 1, pp. 29–43.

[7] L. Typańska-Czajka, The lattice of extensions the relevant logic E , Ph.D Dissertation, Uniwersytet im. Adama Mickiewicza w Poznaniu, 2018.

▸SEBASTIÁN RODRÍGUEZ, Joyal’s completeness theorem: categorical semantics for intermediate logics.

Departamento de Matemáticas, Universidad de Los Andes, Bogotá, Columbia.

E-mail: js.rodriguezl@uniandes.edu.co.

In 1989, Ghilardi gave the first example of an intermediate logic incomplete with respect to its natural Kripke semantics, namely, the logic obtained by adding to intuitionistic (predicative) logic the weak excluded middle and the constant universes axiom schemes. Given this incompleteness phenomenon, with the aim of finding appropriate semantics, we must resort to categorical methods.

A key result is Joyal’s completeness theorem, which states that for any Heyting category $\textsf {H}$ , the evaluation functor $ev : \textsf {H} \to \textsf {Set}^{\textrm {Mod}(\textsf {H})}$ is a conservative Heyting functor. Joyal’s theorem is equivalent to Kripke’s completeness theorem and, if we restrict ourselves to boolean categories is equivalent to classical completeness.

In this talk, we will discuss how to modify Joyal’s theorem, by translating logical properties into categorical ones, in order to get a completeness theorem for intermediate logics. In particular, we apply this method for logics related to constant universes logic. This is a joint work with Xavier Caicedo.

▸MARIELA RUBIN $^*$ AND CAMILLO FIORE, A natural deduction calculus for ST.

Institute for Research in Philosophy, University of Buenos Aires and CONICET, Argentina.

E-mail: marubin@gmail.com.

The main goal of this work is to present a Fitch’s style Natural Deduction Calculus for the non-transitive logic known as ST. ST, originally presented in [1], is a logic that is inferentially identical to Classical Logic but that differs from it at the meta-inferential level by making transitivity fail. That is, even if $\Gamma \vDash _{ST}\Delta $ (that is, all the premisses in $\Gamma $ logically imply some conclusion in $\Delta $ by ST standards) and $\Delta \vDash _{ST}\Theta $ , it might be the case that $\Gamma \nvDash _{ST}\Theta $ (that is, all the premisses in $\Gamma $ do not logically imply any conclusion in $\Theta $ by ST standards).

ST is usually characterized semantically by using an interpreted language (let it be a propositional language for simplicity) doted with a Strong Kleene negation, conjunction, disjunction and implication, and a set of truth values $\mathcal {V}=\{1,\frac {1}{2},0\}$ , with two subsets $\mathcal {S}=\{1\}$ and $\mathcal {T}=\{1,\frac {1}{2}\}$ . We say an inference $\Gamma \vDash _{ST}\Delta $ is ST-valid if and only if for every valuation v, either $v=(\gamma )\in \{\frac {1}{2},0\}$ for some $\gamma \in \Gamma $ or $v(\delta )=1$ for some $\delta \in \Delta $ . With this definition in mind, it is easy to see why ST is inferentially identical to Classical Logic but transitivity fails. There will be no inference $\Gamma \vDash \Delta $ and no valuation v able to make it classically valid but ST-invalid. Yet, it is easy to spot a counterexample to transitivity, i.e., $\gamma \vDash _{ST}\delta $ , $\delta \vDash _{ST}\epsilon $ but $\gamma \nvDash _{ST}\epsilon $ . For there is a valuation v such that $v(\gamma )=1, v(\delta )=\frac {1}{2}$ and $v(\epsilon )=0$ .

At the same time, ST can be characterized by the very well-known sequent calculus introduced by [3], LK minus the Cut-rule. Also known as LK $^-$ . But for the time being there is still no Fitch-style Natural Deduction Calculus for this logic. There are good reasons for wanting this. First, because it is a technical tool that is still absent in the literature. And second, because Natural Deduction Calculi are philosophically interesting in their own right. Following an inferentialist stance, the meaning of logical terms is determined by their inferential role. So semantics in inferentialist terms are defined in terms of a calculus. Yet, for some philosophers such as [2], this is not all. The meaning of a connective is given by the rules that explain how to infer sentences from other sentences, and that is what Fitch-style Natural Deduction Calculus offers: it gives one the information of when we can infer a sentence with some logical term as its main connective and what we can infer from that sentence.

In this presentation we will present the logic, the calculus and a proof that it is both inferentially and meta-inferentially sound and complete with respect to the semantic presentation.

[1] P. Cobreros, P. Égré, D. Ripley, and R. van Rooij, Reaching transparent truth. Mind , vol. 122 (2013), no. 488, pp. 841–866.

[2] M. Dummett, The Logical Basis of Metaphysics , Harvard University Press, Cambridge, 1991.

[3] G. Gentzen, Untersuchungen über das logische Schließen. II. Mathematische Zeitschrift , vol. 39 (1935), no. 1, pp. 405–431.

▸ATTICUS STONESTROM, On f-generic types in NIP groups.

Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA.

E-mail: astones3@nd.edu.

Recall that a definable group is said to be ‘definably amenable’ if it admits a Keisler measure invariant under left-translation. Definable amenability has proved a key hypothesis in much of the generalization of stable group theory to the more general setting of NIP groups, as elucidated over the course of the papers [3], [4], and [2]. An important role in this structure theory is played by “f-generic” types, which serve as the appropriate analogs of “generic” types from the stable setting.

I will define and discuss these notions, and then present a technical result from [5]: for an NIP group, definable amenability is equivalent to the existence of an f-generic type. This answers a question from [2]. As a quick application, I will show that every dp-minimal group is definably amenable, answering a question from [1].

[1] A. Chernikov, A. Pillay, and P. Simon, External definability and groups in NIP theories. Journal of the London Mathematical Society , vol. 80 (2014), no. 1, pp. 213–240.

[2] A. Chernikov and P. Simon, Definably amenable NIP groups. Journal of the American Mathematical Society , vol. 31 (2018), no. 3, pp. 609–641.

[3] E. Hrushovski, Y. Peterzil, and A. Pillay, Groups, measures, and the NIP. Journal of the American Mathematical Society , vol. 21 (2008), no. 2, pp. 563–596.

[4] E. Hrushovski and A. Pillay, On NIP and invariant measures. Journal of the European Mathematical Society , vol. 13 (2011), no. 4, pp. 1005–1061.

[5] A. Stonestrom, On f-generic types in NIP groups, arXiv:2303.13470.

▸DAVID VALDERRAMA, Behavior of some variants of the splitting number in the Hechler and Dual Hechler model.

Departamento de Matemáticas, Universidad de Los Andes, Bogotá, Colombia.

E-mail: d.valderramah@uniandes.edu.co.

In [1] were introduced, among many other new cardinal characteristics, two variants of the splitting number: the cofinally bisecting number and the bisecting number. In this talk, we will present some consistency results about these cardinals obtained using the Hechler and dual Hechler forcing. In particular, we will give a proof of Con( $\mathfrak{d}<\mathfrak{s}_{1/2}$ ) and Con( $\mathfrak{s}_{1/2}^{\infty } < $ non $(\mathcal {E})$ ), where $\mathcal {E}$ is the $\sigma $ -ideal generated by closed sets of measure zero.

[1] J. Brendle, L. J. Halbeisen, L. D. Klausner, M. Lischka, and S. Shelah, Halfway new cardinal characteristics. Annals of Pure and Applied Logic , vol. 174 (2023), no. 9, paper 103303.

[2] D. Valderrama, Cardinal invariants related to density, arXiv 2401.09649, submitted.

▸MAHAN VAZ $^*$ AND MARCELO ESTEBAN CONIGLIO, Non-deterministic dyadic obligation.

Instituto de Filosofia e Ciências Humanas, Universidade Estadual de Campinas, Brazil.

E-mail: mahanvaz@gmail.com.

One of the finest examples of a deontic paradox is Chisholm’s Paradox, which evidences not only the limits of Standard Deontic Logic, but also the intrinsic nature of obligations, that is, obligations can be violated. In such scenarios the argument is usually formalized in a way that also evidences that the standard operators used for obligation and deontic conditional might not be adequate to deal with situations where a primary obligation is violated.

A suggested means to deal with such situations is to introduce either a dyadic obligation operator or non-material conditionals. The first approach has the advantage to changing the way we understand obligation, hence providing new ways of looking into situations where primary obligations are violated. In particular, preference-based logics are usually formalized using such operators [9], bringing with it a notion of betterness, that is, given a situation $\alpha $ , one is obligated to do $\beta $ if in all situations that $\alpha $ occurs, $\beta $ is a better action than any other possible action. Notice also that there is a shift from a normative interpretation of obligation to a praxeological interpretation. Non-material conditional are also an option, because the relation between an obligation and what one must do if they act according to this obligation or contrary to it can be further studied.

Another flank of this work deals with non-deterministic semantics, which stems from the works of Avron and Lev [1, 2]. These kinds of semantics were proposed to deal with logics which could not be semantically characterized by finite matrices and are a perfect fit for logics such as Ivlev’s [10] and paraconsistent logics, for example LFIs [5]. A novel way of treating these kinds of semantics is by means of swap structures first developed by Carnielli and Coniglio [4], which consist in an analytical description of Nmatrices, developed by Avron as mentioned before [3]. These are sets of n-tuples, each coordinate being a bivaluation corresponding to a state of a given formula. For example, given a signature $\Sigma $ which contains a paraconsistent negation $\neg $ and a consistency operator $\circ $ , a set of formulas $For(\Sigma )$ of $\Sigma $ and $\varphi \in For(\Sigma )$ , and a bivaluation $b : For(\Sigma )\to 2$ , then the tuple corresponding to the possible truth-values of $\varphi $ in a swap structure could be $z = (z_1,z_2,z_3)$ , where $z_1 = b(\varphi ), z_2 = b(\neg \varphi ), z_3 = b(\circ \varphi )$ . It is possible then to organize these tuples so that they comply with the usual semantics given for the logics we are working with, thus building models based on such structures. Notice that both the paraconsistent negation $\neg $ and the consistency operator $\circ $ behave non-deterministically in the LFIs.

The proposal of this work is to bring together the non-deterministic approach via swap structures and the study of dyadic obligation operator, giving thus a swap structure semantics for deontic logics with such an operator. We further envisage to develop such semantics together with a paraconsistent approach, incorporating logics such as deontic paraconsistent logics [6, 7, 8] and investigating their behavior via the novel machinery given by swap structures.

[1] A. Avron, Non-deterministic matrices and modular semantics of rules, Logica Universalis (J.-Y. Beziau, editor), Birkhäuser, Basel, 2007, pp. 155–173.

[2] A. Avron and I. Lev, Non-deterministic multiple-valued structures. Journal of Logic and Computation , vol. 15 (2005), pp. 241–261.

[3] A. Avron and A. Zamansky, Non-deterministic semantics for logical systems, Handbook of Philosophical Logic, vol. 16 , (D. M. Gabbay and F. Guenthner, editors), Springer, Netherlands, 2010, pp. 227–304.

[4] W. Carnielli and M. E. Coniglio, Paraconsistent Logic: Consistency, Contradiction and Negation , Logic, Epistemology, and the Unity of Science, 40, Springer, 2016.

[5] W. Carnielli, M. E. Coniglio, and J. Marcos, Logics of Formal Inconsistency, Handbook of Philosophical Logic, vol. 14 , (D. M. Gabbay and F. Guenthner, editors), Springer, Netherlands, 2007, pp. 1–93.

[6] M. E. Coniglio, Logics of deontic inconsistency. Revista Brasileira de Filosofia , vol. 233 (2009), pp. 162–186.

[7] M. E. Coniglio and N. M. Peron, A paraconsistentist approach to Chisholm’s paradox. Principia: An International Journal of Epistemology , vol. 13 (2009), pp. 299–326.

[8] N. C. A. Costa and W. A. Carnielli, On paraconsistent deontic logic. Philosophia , vol. 16 (1986), no. 3-4, pp. 293–305.

[9] L. Goble, Preference semantics for deontic logic part I—Simple models. Logique et Analyse , vol. 46 (2003), no. 183/184, pp. 383–418.

[10] J. Ivlev, A semantics for modal calculi. Bulletin of the Section of Logic , vol. 17 (1988), no. 3-4, pp. 114–121.

Abstracts of talks presented by title

▸ANDRÉS F. URIBE-ZAPATA, Singular values in Cichoń’s diagram: a general theory of iterated forcing using finitely additive measures.

Institute of Discrete Mathematics and Geometry, Faculty of Mathematics and Geoinformation, TU Wien, Vienna, Austria.

E-mail: andres.zapata@tuwien.ac.at.

Saharon Shelah in [2] introduced a finite-support iteration using finitely additive measures to prove that, consistently, the covering of the null ideal may have countable cofinality. Later, in 2019, Jakob Kellner, Saharon Shelah, and Anda R. Tănasie in [1] achieved some new results and applications using such iterations

In this talk, based on the works mentioned above, we present a general theory of iterated forcing using finitely additive measures, which was developed in the speaker’s master’s thesis [3]. For this purpose, we introduce two new notions: on the one hand, we define a new linkedness property, which we call “ $\mu $ - $\mathrm {FAM}$ -linkedness” and, on the other hand, we generalize the idea of intersection number to forcing notions [4], which justifies the limit steps of our iteration theory. Finally, we use this theory to get a new separation of the left side of Cichoń?s diagram allowing a singular value.

This is a joint work with Miguel A. Cardona and Diego A. Mejía.

[1] J. Kellner, S. Shelah, and A. R. Tănasie, Another ordering of the ten cardinal characteristics in Cichoń’s diagram. Commentationes Mathematicae Universitatis Carolinae , vol. 60 (2019), no. 1, pp. 61–95.

[2] S. Shelah, Covering of the null ideal may have countable cofinality. Fundamenta Mathematicae , vol. 166 (2000), no. 1-2, pp. 109–136.

[3] A. Uribe-Zapata, Iterated forcing with finitely additive measures: applications of probability to forcing theory , Doctoral dissertation, Universidad Nacional de Colombia, sede Medellín, 2023.

[4] A. F. Uribe-Zapata, The intersection number for forcing notions, Kyōto Daigaku Sūrikaiseki Kenkyūsho Kōkyūroku , to appear.